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arXiv:1801.00488

Aronszajn's reproducing kernels and Feynman propagators

Pierre-Cyril Aubin-Frankowski

Published Jan 1, 2018• 0 citations
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Abstract

This study shows how Aronszajn's theory of reproducing kernels can be of use for the construction the Hilbert spaces of quantum theory. We show that the Feynman propagator is an example of a reproducing kernel under a boundedness condition. To every Lagrangian thus corresponds a Hilbert space that does not need to be postulated \emph{a priori}. For the free non-relativistic particle, we justify mathematically the concept of space-time granularity. Reproducing kernels allow for a functional, rather than distributional, description of the Hilbert spaces of quantum theory, including the Fock space.

DOI
10.48550/arxiv.1801.00488

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This study shows how Aronszajn's theory of reproducing kernels can be of use for the construction the Hilbert spaces of quantum theory. We show that the Feynman propagator is an example of a reproducing kernel under a boundedness condition. To every Lagrangian thus corresponds a Hilbert space that does not need to be postulated \emph{a priori}. For the free non-relativistic particle, we justify mathematically the concept of space-time granularity. Reproducing kernels allow for a functional, rather than distributional, description of the Hilbert spaces of quantum theory, including the Fock space.
A rigorous development of the continuous representation of Hilbert space by bounded, continuous, multidimensional phase-space functions ψ(p, q) is presented. It is shown that these functions form a closed subspace of L2(p, q) whose elements are functions and not equivalence classes. Differential properties are investigated and it is pointed out that there are a multitude of definitions whereby ψ(p, q) possesses continuous derivatives of all orders. In one of these definitions, each ψ(p, q) is proportional to a multidimensional, entire function f(q − ip), establishing a connection between Bargmann's Hilbert space of entire functions and one example of a continuous representation. Attention is devoted to the purely functional characterization of the continuous representation by means of the reproducing kernel as a special case of Aronszajn's general theory. Properties of various operators in a continuous representation are carefully defined.
In order to preserve the leading role of the action principle in formulating all field theories one needs quantum field theory, with the associated BRST symmetry, and Feynman–DeWitt–Faddeev–Popov ghost fields. Such fields result from the fiber-bundle structure of the space of histories, but the physics-oriented literature used them formally because a rigorous theory of measure and integration was lacking. Motivated by this framework, this paper exploits the previous work of Gill and Zachary, where the use of Banach spaces for the Feynman integral was proposed. The Henstock–Kurzweil integral is first introduced, because it makes it possible to integrate functions like [Formula: see text]. The Lebesgue measure on [Formula: see text] is then built and used to define the measure on every separable Hilbert space. The subsequent step is the construction of a new Hilbert space [Formula: see text], which contains [Formula: see text] as a continuous dense embedding, and contains both the test functions [Formula: see text] and their dual [Formula: see text], the Schwartz space of distributions, as continuous embeddings. This space allows us to construct the Feynman path integral in a manner that maintains its intuitive and computational advantages. We also extend this space to [Formula: see text], where [Formula: see text] is any separable Hilbert space. Last, the existence of a unique universal definition of time, [Formula: see text], that we call historical time, is proven. We use [Formula: see text] as the order parameter for our construction of Feynman’s time ordered operator calculus, which in turn is used to extend the path integral in order to include all time-dependent groups and semigroups with a reproducing kernel representation.
The Feynman propagator used in the conventional in-out formalism in quantum field theory is not a causal propagator as wave packets are propagated virtually instantaneously outside the causal region of the initial state. We formulate a causal in-out formalism in quantum field theory by making use of the Wheeler propagator, the time ordered commutator propagator, which is manifestly causal. Only free scalar field theories and their first quantization are considered. We identify the real Klein Gordon field itself as the wave function of a neutral spinless relativistic particle. Furthermore, we derive a probability density for our relativistic wave packet using the inner product between states that live on a suitably defined Hilbert space of real quantum fields. We show that the time evolution of our probability density is governed by the Wheeler propagator, such that it behaves causally too.
The purpose of this note is to present a "generalized quantum mechanics" in which the number of particles is not given a priori in each problem, and, in particular, is not a constant of the motion. The quantum-mechanical Hilbert space thus contains states with 0, 1, 2,... particles. The interaction picture and the SSS matrix are introduced as usual. It is shown (assuming relativistic invariance) that the calculation of SSS-matrix elements can be performed by means of graphical rules identical to those of Feynman.
Functional quantum theory of free Fermi fields is treated for the special case of a free Dirac field. All other cases run on the same pattern. Starting with the Schwinger functionals of the free Dirac field, functional equations and corresponding many particle functionals can be derived. To establish a functional quantum theory, a physical interpretation of the functionals is required. It is provided by a mapping of the physical Hilbert space into an appropriate functional Hilbert space, which is introduced here. Mathematical details, especially the problems connected with anticommuting functional sources are treated in the appendices.